117 



lerefore th;it there are really in all points Psuch axes parallel tot he 

 IK supposed in S, and that all have the same period. 

 \\ith ivspect to the possible combinations of symmetry-*-!- n 



space-lattices, we can refer here to the contents of the prece- 

 chapters II to IV; the general rules stated there are valid 

 also here. The only question yet to be considered is: what can be 

 K periods of the axes of symmetry in such space-lattices? 

 Let P (fig. 104) be a point of 

 the system, Let us suppose that a 

 symmetry-axis A() of the period 



passes through P, and that it 



perpendicular to the plane 

 )f the figure. According to the 

 ibove, it is therefore at the same 

 time a net-plane of the space- 



ittice. The point situated nearest 

 P in this net-plane may be N v 



icn we perform now the characteristic rotations round A through 

 ingles a, 2, 3#, etc., the point N l reaches successively the corres- 



>nding points N 2 , N 3 , N t , etc. of the net-plane. But because of the 

 )arallellogram-shaped meshes of this net-plane, a point Q must also be 

 found in the net-plane in such a way that Q, N lt N z , and N 3 together 

 form a primary mesh of it. Moreover the coordinates of all these 



)ints in the net-plane must be in rational proportions to each other. 



Now we have supposed that N l was nearest to P ; the absolute 

 listance PQ may therefore only be greater, or in the extremest case 



equal to PN lt etc. Now is evidently ==1 4 shW-j; and 



we calculate the values of this expression for n = 3, 4, 5, 6, etc., 

 /e obtain the following result : x ) 



') The number 2 is of course valid here, as can immediately be seen from a 

 simple figure. 



